Let $(X,\mathcal{X})$ be a standard Borel space, let $\mu$ be a probability measure on $(X,\mathcal{X})$, and let $T \colon X \to X$ be a $\mu$-preserving measurable map. Let $[\mathcal{X}]_\mu$ be the quotient of $\mathcal{X}$ by the equivalence relation
$$ A_1 \sim A_2 \ \Leftrightarrow \ \mu(A_1 \triangle A_2)=0 \text{,} $$
equipped with the natural $\sigma$-algebra $\Sigma_\mu := \sigma( A \mapsto \mu(A \cap B) : B \in [\mathcal{X}]_\mu )$.

For each $\varepsilon>0$, $A \in [\mathcal{X}]_\mu$ and $n \in \mathbb{N} \cup \{0\}$, let
$$ Y_n(\varepsilon,A) \ = \ \{B \in [\mathcal{X}]_\mu : \, |\mu(A \cap T^{-n}(B)) - \mu(A)\mu(B)| \geq \varepsilon \}. $$

The following is simply an alternative way of formulating the standard definition of mixing.

**Definition.** We say that *$\mu$ is $T$-mixing* if for all $\varepsilon>0$ and $A \in [\mathcal{X}]_\mu$,
$$ \limsup_{n \to \infty} Y_n(\varepsilon,A) = \emptyset \text{.} $$

Now in any measurable space $(\Omega,\mathcal{F})$, we have that $\limsup_{n \to \infty} E_n = \emptyset$ if and only if every probability measure $\mathbb{P}$ on $(\Omega,\mathcal{F})$ has $\mathbb{P}(E_n) \to 0$ as $n \to \infty$.

Therefore, an equivalent-or-stronger notion of mixing is the following:

**Definition.** We will say that *$\mu$ is summably $T$-mixing* if for every $\varepsilon>0$, $A \in [\mathcal{X}]_\mu$ and every probability measure $P$ on $([\mathcal{X}]_\mu,\Sigma_\mu)$,
$$ \sum_{n=0}^\infty P\!\left(Y_n(\varepsilon,A)\right) < \infty \text{.} $$

> If $\mu$ is $T$-mixing, does it necessarily follow that $T$ is summably $T$-mixing?
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**Motivation.** It seems that in the theory of random dynamical systems, for a skew-product map $\Theta \colon \Omega \times X \to \Omega \times X$ on a standard Borel space $(X,\mathcal{X})$ over an invertible mixing dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$, given a $\mathbb{P}$-projecting $\Theta$-invariant measure $\mu$ with disintegration $(\mu_\omega)_{\omega \in \Omega}$, one sometimes defines a random correlation function $\rho_{g_1,g_2,\mu}(n,\omega)$ for a pair of functions $g_1,g_2 \in L^\infty(\Omega \times X, \mathcal{F} \otimes \mathcal{X}, \mu)$, by
$$  \left| \int_X g_1\!(\omega,x) \, (g_2 \circ \Theta^n)(\omega,x) \, \mu_\omega(dx) - \int_X g_1(\omega,x) \, \mu_\omega(dx) \int_X g_2(\theta^n\omega,y) \, \mu_{\theta^n\omega}(dy) \right|\text{,} $$
which, for each $g_1,g_2$, is well-defined up to $\mathbb{P}$-a.s. equality.

(See e.g. the question https://mathoverflow.net/questions/327938/)

However, if the answer to my question is *no*, then this seems to me like a slightly problematic approach to extending the concept of "correlation functions" from the deterministic-dynamical-systems setting to the random-dynamical-systems setting. The reason is as follows.

Taking the skew product to be simply a direct product $\Theta = \theta \times T$, and the invariant measure a direct product $\mu=\mathbb{P} \otimes \mu_X$, if $\mu_X$ is $T$-mixing then I feel like any appropriate definition of "random correlation" (if one exists) should give us "almost-sure decay of correlations of bounded observables".

But under the above definition, if $\mu_X$ is mixing and not summably mixing, then taking the base to be the Bernoulli shift on $\Omega=[0,1]^\mathbb{Z}$, letting $g_1(\omega,x)=\mathbf{1}_A(x)$ for a set $A$ that fails to have the summability condition for some probability measure $P$ on $[\mathcal{X}]_{\mu_X}$ and some $\varepsilon>0$, we can construct a measurable set $B \subset [0,1] \times X$ such that the law of a random $X$-section of $B$ is precisely $P$, and we can then set $g_2((\omega_n)_{n \in \mathbb{Z}},x)=\mathbf{1}_B(\omega_0,x)$. The Second Borel-Cantelli Lemma then gives that for $\mathbb{P}$-almost all $\omega$, there is an unbounded set of integers $n$ for which $\rho_{g_1,g_2,\mu}(n,\omega) \geq \varepsilon$.

(The construction here is inspired by the answer to my question https://mathoverflow.net/questions/418835/)

**Further thought.** I retract my statement that a negative answer implies that the above definition of "random correlation" is problematic. It's a perfectly good definition, especially seeing as there exist results about almost-sure exponential decay of such random correlations for certain classes of RDS. But the key point is, to be more accurate, that a negative answer would imply that one probably should not define "almost-sure mixing" simply as almost-sure decay of correlations for all bounded observables on $\Omega \times X$. One other approach that I've seen to studying mixing in RDS is simply to consider $\omega$-independent bounded observables $g_1,g_2 \colon X \to \mathbb{R}$; but a potential problem with this as a way of defining mixing is that it respects the Cartesian-product structure of $\Omega \times X$ (as opposed to treating $\Omega \times X$ just as a space equipped with a projection onto $\Omega$), which goes somewhat contrary to the general philosophy of RDS theory.